Yatharth Kumar

Sarfaraz Equbal, Rohit Gurjar, Yatharth Kumar et al.

We study the problem of fairly assigning a set of discrete tasks (or chores) among a set of agents with additive valuations. Each chore is associated with a start and finish time, and each agent can perform at most one chore at any given time. The goal is to find a fair and efficient schedule of the chores, where fairness pertains to satisfying envy-freeness up to one chore (EF1) and efficiency pertains to maximality (i.e., no unallocated chore can be feasibly assigned to any agent). Our main result is a polynomial-time algorithm for computing an EF1 and maximal schedule for two agents under monotone valuations when the conflict constraints constitute an arbitrary interval graph. The algorithm uses a coloring technique in interval graphs that may be of independent interest. For an arbitrary number of agents with identical additive valuations, we show the existence of an EF1 and maximal schedule when the constraints constitute a path graph. This result uses a reduction to the ``cycle-plus-triangles'' theorem. Using different techniques, we provide an efficient algorithm for finding such a schedule when there are four or more agents and the valuations are further assumed to be dichotomous. We also show that stronger fairness and efficiency properties, including envy-freeness up to any chore (EFX) along with maximality and EF1 along with Pareto optimality, cannot be achieved.

Sarfaraz Equbal, Rohit Gurjar, Ayumi Igarashi et al.

We study the fair allocation of indivisible items subject to conflict constraints. In this framework, the items are represented as the vertices of a graph, with edges corresponding to conflicts between pairs of items. Each agent is assigned an independent set of items from the graph. Our goal is to achieve a fair and efficient allocation of these items. Fairness pertains to satisfying envy-freeness up to one item (EF1), while efficiency is defined by maximality, meaning that no unallocated item can be feasibly assigned to any agent. First, we explore the case of two agents. For monotone valuations, we show that a maximal EF1 allocation always exists on any graph. Our existence proof relies on a color-switching technique, which locally modifies a maximal allocation while preserving feasibility and restoring EF1. We further show that such allocations can be computed in pseudopolynomial time in general, and in polynomial time for additive valuations on arbitrary graphs, as well as for monotone valuations on interval and bipartite graphs. By contrast, once monotonicity is dropped, maximal EF1 allocations need not exist even for identical additive valuations, and deciding existence becomes NP-hard. Next, we consider the case with a general number of agents. Again, we arrive at a negative result: An EF1 and maximal allocation fails to exist even for three agents under identical monotone valuations, and determining the existence of such an allocation is NP-hard. On the positive side, we show that under identical non-monotone additive valuations on a path graph, an EF[1,1] and maximal allocation always exists. This result involves a novel application of the "cycle plus triangles" theorem.